New weighted Poincaré-type inequalities for differential forms (Q2576038)

We first prove local weighted Poincaré-type inequalities for differential forms. Then, by using the local results, we prove global weighted Poincaré-type inequalities for differential forms in John domains, which can be considered as generalizations of the classical Poincaré-type inequality. A weight \(wL^1_{\text{loc}}(\mathbb R^n)\) satisfies the \(A_r\)-condition if \(w>0\) and moreover \[ \sup_B\left(\frac 1{|B|}\int_B w\,dx\right)\left(\frac 1{|B|}\int_Bw^{1/(r-1)}\,dx\right)^{r-1} \] for any ball \(B\subset\mathbb R^n\). Rather advanced local and global (on John subdomains of \(\mathbb R^n\)) weighted Poincaré-type estimates are proved for differential forms. We state one of them: \[ \left(\frac 1{|B|}\int_B |u-u_B|^s\,w^t(x)\,dx\right)^{1/s}\leq C(B)^{1/n}\left(\frac1{|B|}\int_B|du|^pw^{tp/s}(x)\,dx\right)^p, \] where \(u\in {\mathcal D}'(B,\Lambda^1)\) with \(du\in L^p(B,\Lambda^{l+1})\) are differentlal forms with distributional coefficients. Moreover \(1<p<n\); \(l=1,\dots,n\); \(0<t< 1/r(1-1/p+1/n)\); \(s =np(1-tr)/(n-p)\) and \(B\subset \mathbb R^n\) is any ball or cube, \(u_B\in{\mathcal D}'(B,\Lambda^1)\) is a certain average of the form \(u\).